#8429332

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10,15,20)(11,14,19,16)(12,18,13,17)(21,24,22,23)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,19)(10,16,20,14)(12,21,13,22)(17,24,18,23)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3,5,4)(2,6,7,8)(9,12,15,13)(10,17,20,18)(11,21,19,22)(14,23,16,24)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,13,15,12)(10,18,20,17)(11,22,19,21)(14,24,16,23);(1,4,5,3)(2,6,7,8)(9,14,15,16)(10,19,20,11)(12,23,13,24)(17,22,18,21)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336528);